Error variance estimation in semi-functional partially linear regression models

Abstract

This paper focuses on partially linear regression models with several real and functional covariates. The aim is to construct an estimate of the variance of the error. In our model, a real-valued response variable is explained by the sum of an unknown linear combination of the components of a multivariate random variable and an unknown transformation of a functional random variable, and the second sample moment based on residuals from a semiparametric fit is proposed for estimating the error variance. Then, the asymptotic normality and the law of the iterated logarithm of such estimator are obtained. Finally, a simulation study illustrates the finite sample behaviour of the estimator, while an application to real data shows the usefulness of the proposed methodology, more specifically for confidence region construction.

Keywords:

• functional data
• semi-functional partial linear regression
• error variance estimation
• asymptotic normality
• confidence regions
• law of the iterated logarithm

AMS Subject Classifications:

• 62G05
• 62F12

Acknowledgements

The authors wish to thank the Editor-in-Chief, Associate Editor and two anonymous referees for their helpful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

9. Funding

The research of G. Aneiros and P. Vieu was partly supported by Spanish Ministerio de Economía y Competitividad [grant number MTM2014-52876-R]. Ling's work is supported by the National Social Science Fund of China (14ATJ005) and the NNSF of China (11171001).